Identification of critical nodes in large-scale spatial networks

Vishaal Krishnan and Sonia Martínez
IEEE Transactions on Control of Network Systems, 29 (3) (2021) 1246-1256

Abstract:

In this work, we address a problem of identifying critical nodes in large-scale spatial networks. We adopt a notion of criticality related to the smallest positive eigenvalue of the graph Laplacian, which governs the rate of convergence of the Laplacian consensus dynamics. We consider spatial networks in which the nodes are uniformly distributed over a spatial region and communicate with their local spatial neighbors. Through a continuum approximation, we formulate the consensus problem by means of a partial differential equation involving the Laplace operator, subject to the Neumann boundary condition. As with its finite-dimensional counterpart, we observehow the performance of these dynamics is directly related to the second smallest eigenvalue of the Laplace operator over the domain of interest. We then reduce the critical node set identification problem to that of finding a ball of fixed radius, whose removal minimizes the rate of convergence over the residual domain. This leads us to consider two functional optimization problems. First, we treat the problem of determining the second smallest eigenvalue for a fixed domain by minimizing an energy functional. We characterize the critical points of the energy functional, and then construct the gradient dynamics that converge to the set of critical points. We then prove that the only locally asymptotically stable critical point is the second eigenfunction of the Laplace operator. Building on these results, we consider the critical ball identification problem, provide a characterization of the critical points, and define gradient dynamics to converge asymptotically to these points. Finally, for the case of clusters of small size, we present an approximate identification method.

File: (ArXiv version)

Bib-tex entry:

@article{VK-SM:21-tcns,
author = {V. Krishnan and S. Mart{\'\i}nez},
title = {Identification of critical nodes in large-scale spatial networks},
journal= {IEEE Transactions on Control of Network Systems},
year = {2021},
pages = {1246--1256},
volume = {29},
number = {3}
}